A Combined Fourier Series–Galerkin Method for the Analysis of Functionally Graded Beams

2004 ◽  
Vol 71 (3) ◽  
pp. 421-424 ◽  
Author(s):  
H. Zhu ◽  
B. V. Sankar
Keyword(s):  
Fourier Series ◽  
Differential Equations ◽  
Galerkin Method ◽  
Functionally Graded ◽  
Shear Stresses ◽  
Functionally Graded Beam ◽  
Fourier Series Method ◽  
Bending Stresses ◽  
The Galerkin Method ◽  
Graded Structures

The method of Fourier analysis is combined with the Galerkin method for solving the two-dimensional elasticity equations for a functionally graded beam subjected to transverse loads. The variation of the Young’s modulus through the thickness is given by a polynomial in the thickness coordinate and the Poisson’s ratio is assumed to be constant. The Fourier series method is used to reduce the partial differential equations to a pair of ordinary differential equations, which are solved using the Galerkin method. Results for bending stresses and transverse shear stresses in various beams show excellent agreement with available exact solutions. The method will be useful in analyzing functionally graded structures with arbitrary variation of properties.

Dominance of Shear Stresses in Early Stages of Impulsive Motion of Beams

1960 ◽  
Vol 27 (1) ◽  
pp. 132-138 ◽  
Author(s):  
H. H. Bleich ◽  
R. Shaw
Keyword(s):  
Differential Equations ◽  
Timoshenko Beam ◽  
Shear Stresses ◽  
Impulsive Motion ◽  
Early Stages ◽  
Plastic Analysis ◽  
Bending Stresses ◽  
Impulsive Forces

In order to compare the magnitude of bending stresses and shear stresses in beams under the action of impulsive forces, the values of these stresses are determined from the known differential equations for the Timoshenko beam. It is found that in the early stages, soon after the initiation of the motion, the shear stresses are of much larger magnitude than the bending stresses. This result indicates that for sufficiently large initial velocities first yielding will be in shear, a matter of consequence in plastic analysis.

A Theoretical Series Solution for the Dynamics of Finite-Length Bearings and Squeeze-Film Dampers

2003 ◽  
Author(s):  
Jose´ Antunes ◽  
Miguel Moreira ◽  
Philippe Piteau
Keyword(s):  
Fourier Series ◽  
Galerkin Method ◽  
Finite Length ◽  
Reynolds Equation ◽  
Double Fourier Series ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Squeeze Film Dampers ◽  
Galerkin Solution ◽  
The Galerkin Method

In this paper we develop a non-linear dynamical solution for finite length bearings and squeeze-film dampers based on a Spectral-Galerkin method. In this approach the gap-averaged pressure is approximated, in the lubrication Reynolds equation, by a truncated double Fourier series. The Galerkin method, applied over the residuals so obtained, generate a set of simultaneous algebraic equations for the time-dependent coefficients of the double Fourier series for the pressure. In order to assert the validity of our 2D–Spectral-Galerkin solution we present some preliminary comparative numerical simulations, which display satisfactory results up to eccentricities of about 0.9 of the reduced fluid gap H/R. The so-called long and short-bearing dynamical solutions of the Reynolds equation, reformulated in Cartesian coordinates, are also presented and compared with the corresponding classic solutions found on literature.

scholarly journals Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 999
Author(s):  
Dana Černá
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Jump Diffusion ◽  
Diffusion Models ◽  
Dirichlet Boundary Conditions ◽  
Quadratic Spline ◽  
Vanishing Moments ◽  
Bounded Interval ◽  
Spline Wavelets ◽  
The Galerkin Method

This paper is concerned with a construction of new quadratic spline wavelets on a bounded interval satisfying homogeneous Dirichlet boundary conditions. The inner wavelets are translations and dilations of four generators. Two of them are symmetrical and two anti-symmetrical. The wavelets have three vanishing moments and the basis is well-conditioned. Furthermore, wavelets at levels i and j where i - j > 2 are orthogonal. Thus, matrices arising from discretization by the Galerkin method with this basis have O 1 nonzero entries in each column for various types of differential equations, which is not the case for most other wavelet bases. To illustrate applicability, the constructed bases are used for option pricing under jump–diffusion models, which are represented by partial integro-differential equations. Due to the orthogonality property and decay of entries of matrices corresponding to the integral term, the Crank–Nicolson method with Richardson extrapolation combined with the wavelet–Galerkin method also leads to matrices that can be approximated by matrices with O 1 nonzero entries in each column. Numerical experiments are provided for European options under the Merton model.

scholarly journals An Efficient Beam Element Based on Quasi-3D Theory for Static Bending Analysis of Functionally Graded Beams

Materials ◽  
2019 ◽  
Vol 12 (13) ◽  
pp. 2198 ◽  
Author(s):  
Hoang Nam Nguyen ◽  
Tran Thi Hong ◽  
Pham Van Vinh ◽  
Do Van Thom
Keyword(s):  
Transverse Shear ◽  
Volume Fraction ◽  
Beam Theory ◽  
Beam Element ◽  
Functionally Graded ◽  
Shear Stresses ◽  
Functionally Graded Beam ◽  
Transverse Shear Stresses ◽  
Power Law Index ◽  
Transverse Shear Strains

In this paper, a 2-node beam element is developed based on Quasi-3D beam theory and mixed formulation for static bending of functionally graded (FG) beams. The transverse shear strains and stresses of the proposed beam element are parabolic distributions through the thickness of the beam and the transverse shear stresses on the top and bottom surfaces of the beam vanish. The proposed beam element is free of shear-looking without selective or reduced integration. The material properties of the functionally graded beam are assumed to vary according to the power-law index of the volume fraction of the constituents through the thickness of the beam. The numerical results of this study are compared with published results to illustrate the accuracy and convenience rate of the new beam element. The influence of some parametrics on the bending behavior of FGM beams is investigated.

scholarly journals Solving nonlinear boundary value problems by the Galerkin method with sinc functions

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Sertan Alkan ◽  
Aydin Secer
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Sinc Approximation ◽  
Sinc Functions ◽  
The Galerkin Method

AbstractIn this paper, the sinc-Galerkin method is used for numerically solving a class of nonlinear differential equations with boundary conditions. The importance of this study is that sinc approximation of the nonlinear term is stated as a new theorem. The method introduced here is tested on some nonlinear problems and is shown to be a very efficient and powerful tool for obtaining approximate solutions of nonlinear ordinary differential equations.

Dynamics of a Cantilever Beam With Attached Pendulum

2001 ◽  
Author(s):  
Danuta Sado
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Cantilever Beam ◽  
Weakly Nonlinear ◽  
Second Mode ◽  
Linear Viscoelastic ◽  
Nonlinear Pendulum ◽  
Analysis Of Dynamics ◽  
The Galerkin Method ◽  
Euler Bernoulli Beam

Abstract This work draws attention to the to the analysis of dynamics of a nonlinear coupled cantilever beam-pendulum oscillator. Dynamical systems of this type have important technical applications, because many mechanical components consist of linear or weakly nonlinear continuos substructures such as beam coupled to nonlinear oscillators. The present paper is a continuation of the author’s previous work where in applying the Galerkin method the modal series was truncated at the first mode. In this work it is assumed that the cantilever beam behaves like an Euler-Bernoulli beam and to its end pendulum is attached. The integro-differential equations are transformed into an ordinary differential equations with the use of Galerkin procedure with beam functions. In this study, in applying the Galerkin method the modal series was truncated at the second mode. Next these equations were solved numerically and there was studied the effect of the internal friction on energy transfer in a coupled structure that consist of a linear viscoelastic beam supporting at its tip a nonlinear pendulum.

Steady-state analysis of DC converter using Galerkin’s method

2019 ◽  
Vol 38 (6) ◽  
pp. 2057-2069
Author(s):  
Igor Korotyeyev
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Galerkin Method ◽  
State Space ◽  
Time Interval ◽  
Time Varying ◽  
Content Type ◽  
State Process ◽  
The Galerkin Method ◽  
Steady State Process

Purpose The purpose of this paper is to present the Galerkin method for analysis of steady-state processes in periodically time-varying circuits. Design/methodology/approach A converter circuit working on a time-varying load is often controlled by different signals. In the case of incommensurable frequencies, one can find a steady-state process only via calculation of a transient process. As the obtained results will not be periodical, one must repeat this procedure to calculate the steady-state process on a different time interval. The proposed methodology is based on the expansion of ordinary differential equations with one time variable into a domain of two independent variables of time. In this case, the steady-state process will be periodical. This process is calculated by the use of the Galerkin method with bases and weight functions in the form of the double Fourier series. Findings Expansion of differential equations and use of the Galerkin method enable discovery of the steady-state processes in converter circuits. Steady-state processes in the circuits of buck and boost converters are calculated and results are compared with numerical and generalized state-space averaging methods. Originality/value The Galerkin method is used to find a steady-state process in a converter circuit with a time-varying load. Processes in such a load depend on two incommensurable signals. The state-space averaging method is generalized for extended differential equations. A balance of active power for extended equations is shown.

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